Theorem nnssn0s 28391

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28385	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4089	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3982	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3901   ⊆ wss 3904  {csn 4581   0_s c0s 27875  ℕ_0scn0s 28382  ℕ_scnns 28383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-ss 3921  df-nns 28385

This theorem is referenced by:  nnssno  28392  nnn0s  28397  nnn0sd  28398  nnsgt0  28409